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Orthogonal duality of toric Fano varieties with regular involution. (English. Russian original) Zbl 1116.14045
Russ. Math. Surv. 61, No. 3, 553-554 (2006); translation from Usp. Mat. Nauk 61, No. 3, 165-166 (2006).
From the text: According to V. Batyrev [J. Algebr. Geom. 3, No. 3, 493–535 (1994; Zbl 0829.14023)], Gorenstein toric Fano varieties correspond to reflexive polyhedra. The author extends the notion of reflexivity in the following sense: A weight is a positive integer $$m_i$$ for each couple of opposite integral points of the dual polyhedron (hence, reflexive polyhedra correspond to the case of weights 1). When all integer points of the dual polyhedron are vertices of it, the weighted polyhedron corresponds to an ample divisor invariant under an involution on a Fano variety.
A construction for an orthogonal weighted polyhedron is given, and its reflexivity is conjectured. Furthermore, a large class of examples is constructed via graph theory, and a theory of their orthogonal duality is given (the proof is indicated to be similar to Proposition 2.4 in the author’s paper [Proc. Steklov Inst. Math. 246, 3–12 (2004); translation from Tr. Mat. Inst. Steklova 246, 10–19 (2004; Zbl 1116.14029)].
##### MSC:
 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14J25 Special surfaces
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